One-Way Quantum Repeater Based on Near-Deterministic Photon-Emitter Interfaces
Borregaard, Johannes; Pichler, Hannes; Schröder, Tim; Lukin, Mikhail D.; Lodahl, Peter; Sørensen, Anders S. DOI 10.1103/PHYSREVX.10.021071 Publication date 2020 Document Version Final published version Published in
Physical Review X
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Borregaard, J., Pichler, H., Schröder, T., Lukin, M. D., Lodahl, P., & Sørensen, A. S. (2020). One-Way Quantum Repeater Based on Near-Deterministic Photon-Emitter Interfaces. Physical Review X, 10(2), [021071]. https://doi.org/10.1103/PHYSREVX.10.021071
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One-Way Quantum Repeater Based on Near-Deterministic Photon-Emitter Interfaces
Johannes Borregaard ,1,2Hannes Pichler,3,4Tim Schröder,5,6Mikhail D. Lukin,4Peter Lodahl,6and Anders S. Sørensen61QMATH, Department of Mathematical Sciences, University of Copenhagen,
DK-2100 Copenhagen Ø, Denmark
2QuTech and Kavli Institute of Nanoscience, Delft University of Technology,
Lorentzweg 1, 2628 CJ Delft, The Netherlands
3ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA 4
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
5Department of Physics, Humboldt-Universität, 12489 Berlin, Germany 6
Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark
(Received 12 July 2019; revised manuscript received 15 April 2020; accepted 1 May 2020; published 30 June 2020) We propose a novel one-way quantum repeater architecture based on photonic tree-cluster states.
Encoding a qubit in a photonic tree cluster protects the information from transmission loss and enables long-range quantum communication through a chain of repeater stations. As opposed to conventional approaches that are limited by the two-way communication time, the overall transmission rate of the current quantum repeater protocol is determined by the local processing time enabling very high communication rates. We further show that such a repeater can be constructed with as little as two stationary qubits and one quantum emitter per repeater station, which significantly increases the experimental feasibility. We discuss potential implementations with diamond defect centers and semiconductor quantum dots efficiently coupled to photonic nanostructures and outline how such systems may be integrated into repeater stations. DOI:10.1103/PhysRevX.10.021071 Subject Areas: Quantum Physics,
Quantum Information
I. INTRODUCTION
Encoding information in quantum systems is the funda-mental principle of quantum information technologies, ranging from quantum computers [1] to unconditionally secure communication [2]. Quantum networks constitute an important element for implementing such technologies in a scalable fashion [3]. The exact requirements and applications of large-scale quantum networks constitute an active research area [4]. One of the key challenges for constructing large-scale quantum networks is to faithfully transmit quantum information over long distances, which is challenging due to transmission loss.
Quantum repeaters have been proposed as a means to overcome transmission loss by exploiting quantum corre-lations to extend the transmission length of quantum information [5–7]. The conventional quantum repeater architecture relies on heralded quantum entanglement distribution, which necessitates long-lived quantum memo-ries and two-way communication between sender and
receiver [6]. The need for heralding limits the communi-cation rate at which quantum information can be distributed and requires long-lived quantum memories with efficient light-matter coupling[8,9]. To overcome these limitations, one-way and all-photonic quantum repeaters have been proposed [10–16]. One-way repeaters use multiphoton encoding and quantum error-correcting codes to protect the quantum information from both loss and operational errors. In this way, quantum information can be transmitted from one repeater station to the next without the need for preestablished entangled links. For these reasons, in prin-ciple, one-way repeaters can significantly boost the dis-tribution rate[7]without the need for a long-lived quantum memory for key applications such as long-distance quan-tum key distribution [17]. An outstanding challenge involving the physical implementation of one-way quan-tum repeaters is how to efficiently generate the multiqubit error-correcting codes and how to perform error correction. This usually requires many high-fidelity two-qubit oper-ations and considerable amounts of auxiliary qubits at each repeater station[12,15,16,18].
In this article, we propose a novel one-way quantum repeater architecture that can be implemented with as little as two memory qubits and one quantum emitter per repeater station. Our approach is based on photonic tree-cluster states [19], which are used to encode a message Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
qubit to be transmitted to the next repeater station (see Fig. 1). Photonic tree clusters have previously been considered as photonic memories to ensure efficient entan-glement swapping in all-optical quantum repeaters[14,20]. Such all-optical approaches generate multiple photonic tree clusters at each repeater station potentially requiring kilo-meter-long delay lines and millions of single-photon sources per station[20]. Our approach circumvents this significant overhead by using strongly coupled quantum emitters with built-in nonlinearity. Specifically, in our approach, the photonic tree clusters required for the repeater can be generated with two memory qubits and one single-photon emitter per repeater station using repeated photon emissions
[21]. In addition, correction of losses requires only a single Bell measurement independent of the size of the tree encoding. This constitutes a significant reduction in overhead as compared to, e.g., one-way quantum repeaters based on the quantum parity encoding[11,12,15], which requires a number of two-qubit operations that scale linearly with the size of the encoding corresponding to hundreds of memory qubits per repeater station [7]. In comparison, the current approach can be implemented with only two spin systems per repeater station, as we outline below. We also discuss possible experimental implementations of our protocol based on state-of-the-art solid-state quantum emitters in nano-photonic structures in order to lay out a realistic path toward high bit rate, long-range quantum communication. Importantly, many of the required parameters for our pro-tocol are not far from current state-of-the art performances, which together with the significant resource reduction compared to previous one-way protocols cements the exper-imental feasibility of our approach.
II. QUANTUM REPEATER PROTOCOL The basic operation of the repeater is shown in Fig.1. At each node, a multiphoton entangled state is generated and used to encode and transmit a message qubit to the next repeater station. Crucially, even if some of the photons are lost, the repeater can decode the logical qubit and reencode it, thereby correcting for photon loss before transmitting the message to the next station.
A. Tree-cluster states
In this work, we consider one-way quantum repeaters based on using tree-cluster states as error-correcting codes. Such tree-cluster states are illustrated in Fig.2. We character-ize the tree by a branching vector¯t ¼ ½b0; b1;…; bd, which
specifies the connectivity of the tree as one moves from the root vertex [top node in Fig.2(a)] through the d levels of the tree. Tree-cluster states are obtained by associating a qubit with each vertex (see Fig.1). Moreover, one further asso-ciates a stabilizer operator, Ki¼ σxi ⊗j∈N ðiÞσzj, that acts
nontrivially on the vertex i and its neighborsN ðiÞ. The tree-cluster state is the unique eigenstate with eigenvalueþ1 of all stabilizer operators Ki.
As a specific, illustrative example, let us consider a [2, 2] tree-cluster state [see Fig.2(a)]. One can easily check that this 7-qubit state is given by
jψi ¼ 1
23=2ðj0isðj0ijþ; þi þ j1ij−; −iÞ⊗2
þ j1isðj0ijþ; þi − j1ij−; −iÞ⊗2Þ: ð1Þ
Encoding Reencoding Decoding
FIG. 1. Sketch of a one-way quantum repeater with photonic tree clusters. A message qubit (orange dot) is encoded through a Bell measurement (dashed box) with the root spin qubit of a photonic tree cluster. An example with a [2,3,2] tree is shown where dots correspond to photonic qubits and solid lines indicate correlations. The encoded qubit (depicted in the orange boxes) is sent a distance L0to the next repeater station. At the next repeater station, the qubit information is reencoded through a Bell measurement between one of the first-level photonic qubits and the root spin qubit of a new tree. The remaining qubits of the incoming tree are measured with single qubit z (red circle) or x (orange circle) measurements. The reencoding can succeed despite multiple photons being lost in transmission (gray dots and dashed lines). At the end station, the message qubit can be transferred to, e.g., a receiving spin qubit by means of a controlled-phase (CZ) gate.
Here, we have defined the states ji ¼ ðj0i j1iÞ=pffiffiffi2, withj0i and j1i being the basis states of the qubits. Below we consider an implementation where the root qubit is represented by a stationary two-level spin system (spin qubits are denoted with subscript s) while the rest of the tree cluster state is represented by photons.
B. Encoding the logical qubit
Consider the situation where the message qubit, αj0isþ βj1is, is initially prepared in a second stationary
qubit. To send this message qubit from the first station to the next repeater, one has to encode it into the state of the photons. This can be achieved by a simple teleportation process, which can be realized by a 2-qubit Bell measure-ment of the stationary qubits, i.e., the spin that stored the message qubit and the stationary root qubit of the tree-cluster state [see Fig.2(b)].
For the above example of a [2, 2] tree, this prepares the state of the photons in
jΨi ¼ð1 þ x1Þα þ ð1 − x4 1Þβðj0; þ;þi þ j1; −;−iÞ⊗2
þ x2ð1 − x1Þα þ ð1 þ x4 1Þβðj0; þ; þi − j1; −;−iÞ⊗2;
ð2Þ
where x1¼ 1 and x2¼ 1, depending on the four possible outcomes of the Bell measurement. The values of x1and x2are not important since they can eventually be corrected in the decoding step. In Fig. 2, we assume for concreteness that we have obtained the values x1¼ x2¼ 1. Note that the quantum information of the message qubit is stored in a nonlocal form in the photonic degrees of freedom, and cannot be retrieved by observing, e.g., only a single photon.
C. Photon loss and recovery of the message qubit After this encoding step, the photons are transmitted to the next repeater station. The specific encoding protects against transmission loss such that the effective trans-mission probability of the message qubit is significantly increased compared to the bare transmission of a single photon.
To illustrate the basic mechanism, we again consider the example of the [2, 2] encoding in Eq.(2). Already in this simple encoding, one can tolerate the loss of up to two photons in one of the two branches [see Fig.2(c)]. To see this, it is instructive to consider how the state can be recovered and the quantum information retrieved. As a first step in the recovering process, one measures all the qubits in one of the two branches (in Fig.2, we assume that the left branch is to be measured). Specifically, the first-level qubit is measured in the z basis and the second-level qubits are measured in the x basis. Note that the corresponding measurement outcomes are perfectly correlated, such that only two sets of outcomes for the three measurements are possible. This is crucial, as it allows us to infer the outcome from each of the three measurements, even if two of those qubits are lost. This measurements projects the state of the qubits in the remaining branch [right branch in Fig2(c)] into the state
ð1 þ x1Þα þ ð1 − x1Þβ
2pffiffiffi2 ðj0; þ; þi þ j1; −; −iÞ þ x2x3ð1 − x1Þα þ ð1 þ x1Þβ
2pffiffiffi2 ðj0; þ; þi − j1; −; −iÞ; ð3Þ where x3¼ 1, depending on the measurement outcome. This branch now contains the entire encoded quantum information. It can be retrieved by measuring the two second-level qubits in the z basis [see Fig.2(d)]. Simple algebra shows that this prepares the remaining first-level qubit in the state αj0i þ βj1i (up to known Pauli correc-tions that only depend on the obtained measurement outcomes xi). This simple analysis shows that the retrieval
of the message qubit from a [2, 2] tree is possible as long as one branch is not corrupted and not more than two qubits of the other branch are lost, illustrating the basic principle allowing for correction of photon loss. Increasing the tree
(a) (b)
(c) (d)
FIG. 2. Sketch of the principle behind counterfactual error correction in the tree-cluster encoding. A [2, 2] tree is considered for simplicity (a). A qubitαj0i þ βj1i is encoded through a Bell measurement between this qubit and the root spin qubit (b). Loss of qubits (gray vertices) from one branch (c) still allows retrieving the encoded information by measuring at least one remaining second-level qubit in the x basis and the second-level qubits of the intact branch in the z basis (d) despite the fact that we do not get any information from (attempted) measurements on the lost photons (dashed circles).
depth (length of¯t) and the number of branches increases the robustness of the encoding by the same principle [19].
D. Reencoding and repetition
The goal of the repeater station is to reencode the retrieved qubit in a new tree. This can be achieved in complete analogy to the encoding of the message qubit at the sending station: first a new tree cluster of photons is generated, with a stationary spin serving as the root qubit (see Sec.III A), followed by a Bell measurement between the message qubit and the root qubit.
Above we described how to recover the qubit into a photonic qubit in the highest level of the tree. In the reencoding procedure the goal is instead to perform a Bell measurement between the encoded qubit and the root qubit of a new tree. In analogy with the procedure above, this reencoding simply requires a Bell measurement between one of the first-level photons of the encoded tree cluster and the root qubit along with measurement of all other qubits in the same bases as above. Note that the order of the measurements is not important in the above recovery scheme. In practice, this allows us to reencode the quantum information at each repeater station without prior knowl-edge about which qubit was lost. Specifically, one can first attempt a Bell measurement between one of the first-level qubits and the root qubit of the new tree. If this measure-ment is successful, one can teleport the encoded quantum information into the new tree by measuring all connected qubits in the z basis. Some of these measurements may turn out to be unsuccessful because the qubits were lost in transmission. In these cases, the corresponding measure-ment outcome is inferred through measuremeasure-ments on qubits in the next level of the corresponding branch, in complete analogy to the example above. If the first Bell measurement is unsuccessful itself (because the corresponding photon was lost in transmission), then the value of a z measurement can be inferred instead (via measurement of the next-level qubits), and a Bell measurement can be attempted with another first-level qubit. In order not to perturb the root qubit of the new tree in a failed attempt of a Bell measurement, special care must be given to the imple-mentation of the measurement, as described below. Specifically, the message qubit should first be transferred to an auxiliary spin qubit by means of a spin-photon controlled-phase gate (CZgate) and then encoded into the new tree cluster with a deterministic Bell measurement between the auxiliary spin qubit and the root spin qubit (see Fig. 4).
The reencoding and transmission continues down the repeater chain until the encoded message qubit arrives at the end node. There the message qubit can be either transferred to a stationary spin in a similar fashion as in the repetition step (see Fig.1) or directly measured (without first transferring the information to a receiving spin qubit)
by appropriate measurements of the photons of the encoded tree.
III. EXPERIMENTAL IMPLEMENTATION The key requirements for an implementation of the above protocol are the ability to generate tree-cluster states of photons, realize Bell measurements between stationary spins and photonic qubits, and perform measurements of the photons in the x and z basis.
A. Photonic tree generation
We propose to generate the photonic tree cluster states using a light-matter interface illustrated in Fig.4. It consists of stationary memory spins and one spin which is coupled to the light field. The latter is used to generate photons by selectively coupling a ground statej1is to an excitedjeis via the optical field in a one-sided cavity. We are consid-ering a time-bin representation of the photonic qubits. In this representation, the presence of a single photon in one of two nonoverlapping spatiotemporal modes represents the two qubit statesj0i and j1i. The main reason for using this time-bin representation is that it allows us to detect errors stemming from photon loss. This could also be obtained with a polarization representation, but the time-bin repre-sentation is better suited for transmittance through optical fibers, which typically disturb the polarization state.
Recent work [21] showed that sequential excitation of the quantum emitter, together with controlled-phase gates between the emitter and the memory spins, allows us to deterministically generate an arbitrary photonic tree-cluster state. In particular, a tree-cluster state of depth dþ 1 requires only d memory spin systems while the number of necessary spin-spin CZ gates scales polynomially with
the number of branches at each level[21]. In what follows, we show that tree-cluster states of depth 3 are sufficient for transmission distances up to 1000 km (assuming telecom frequencies) and consequently only two memory spin systems and a single quantum emitter are necessary for the generation of such states.
The generation of a tree-cluster state with depth 3 is sketched in Fig.3. In the first step [Fig.3(a)],CZgates are
applied between the two memory spins and between the emitter and one memory spin. The spin of the quantum emitter acts as a second-level qubit in the tree, and all third-level photonic qubits of the subbranch are emitted through repeated excitation followed by spontaneous emission, as shown in Fig.3(b). The second-level qubit is then mapped to a photonic qubit using the auxiliary level of the emitter, which also detaches the emitter from the preliminary tree-cluster state. This step is repeated until all second- and third-level photons of the branch have been emitted. The spin of the second memory qubit is then first swapped to the spin of the emitter by means of two CZ gates and
This completes the emission of one branch of the photonic tree-cluster state, and the procedure can then be repeated to output the entire state. We note that this generation procedure, in principle, requires additional Hadamard gates on the third-level photons to fit the stabilizer description introduced earlier. This is, however, not necessary since it is sufficient to simply rotate the measurement basis of these photons in the reencoding and decoding steps.
A drawback of this generation scheme is that the photons will be emitted such that the first-level qubit of a branch is emitted last. As previously described, the presence or absence of a first-level photon determines the measurement basis of the corresponding branch at a repeater station. It is therefore necessary to delay the photons of each branch to enable measuring the first-level qubit first. The length of this delay will depend on the emission rate of the emitters and the number of photons per branch, but is generally modest and implementable in optical fiber delays. We discuss this issue in more detail below.
B. Bell measurement
As described above, the reencoding at the repeater stations requires a successful Bell measurement between
one of the first-level qubits in the encoded tree cluster and the root qubit of the new tree cluster. It is crucial that this Bell measurement is designed with a built-in error detection: if a Bell measurement is attempted with a lost first-level qubit, the measurement should abort without perturbing the root qubit. Otherwise, a new tree cluster has to be generated after each failed attempt.
The setup required to generate the tree-cluster states [Fig.3(a)] conveniently also allows for such an operation. While one of the stationary memory qubits represents the root qubit, the spin coupled to a one-sided cavity is used for heralded storage of the message qubit through a spin-photon CZ gate (see below) [22–25]. Importantly, the
success of the storage is conditioned on subsequently CZ CZ CZ CZ (a) (b) (c)
FIG. 3. Generation of photonic tree-cluster states. Two memory spin systems (S1, S2) and a quantum emitter (E) are all prepared in statejþis, andCZgates entangle the spins as shown in (a). In
the second step (b), all third-level photons (purple dots) of one sub-branch of the tree cluster are emitted (operation C) followed by the emission of the corresponding second-level photon (blue dot), which detaches the emitter from the preliminary tree cluster (operation M). Operations M and C can be implemented using a sequence of π pulses on various transitions in the emitter, as indicated by the color code in (b) referring to the color of the transitions in the emitter level diagram shown in the inset. Here driven transitions are indicated by solid lines, whereas decay (γ) is represented by the wiggle line. To emit more second-level qubits, E is again prepared injþisfollowed by aCZgate with S2
and step (b) is repeated. In step (c), the state of the memory spin is first swapped to the emitter and then emitted as the first-level photon of the branch. Here the emission of the first-level photon is achieved by an operationM, which is similar to M except that the red π pulse and subsequent decay is replaced by a weak driving of the j2is− jeis transition to ensure that the photons have a narrow bandwidth. The steps are then repeated until the entire tree has been emitted.
CZ
FIG. 4. The basic elements of the reencoding operation at the repeater stations. (a) The reencoding is performed in a loss tolerant manner by first performing a heralded storage of the message qubit in an auxiliary memory spin (dashed box). This is obtained through a spin-photon controlled-phase (CZ) gate with a
first-level photon of the encoded tree. The transfer is heralded by the detection of the photon in the x basis. If unsuccessful (photon loss), the auxiliary spin is initialized and the operation is tried again with another first-level photon. Once the storage is successful, a deterministic Bell measurement is performed between the auxiliary spin and the root spin qubit of the new tree cluster, thereby completing the reencoding operation. (b) The experimental implementation of the reencoding operation. The spin-photonCZgate is performed by reflecting the photonic qubit off a one-sided cavity (shown as a photonic crystal cavity) strongly coupled to a quantum emitter, as described in the text. The parameters g,γ, and κ are the single-photon Rabi frequency of the optical transition, the spontaneous emission rate of the emitter, and the decay rate of the cavity field, respectively. The same physical setup is used to generate the tree-cluster state requiring an extra spin qubit and one auxiliary level (j2is) in the quantum emitter (see Fig.3).
detecting the photon in the x basis [see Fig.4(a)]. When a storage attempt is unsuccessful due to the loss of the photonic qubit, the auxiliary spin qubit is simply reinitial-ized and a new attempt is made with another first-level qubit. The root spin qubit of the new tree cluster is completely unaffected by this. Once the storage is suc-cessful, a Bell measurement between the auxiliary spin system and the spin system containing the root qubit of the new tree is performed using deterministic entangling gates between the two spin systems, concluding the reencod-ing step.
To perform the cavity-mediated spin-photonCZgate, we assume the auxiliary spin to initially be prepared in ground statej0is. For a time-bin encoded photonic qubit, the early half of the wave packet corresponding to qubit statej0i is first reflected off the cavity. In the ideal limit, the photon will be reflected with aπ-phase shift from the cavity. Now the transformation j0is→ ðj0isþ j1isÞ=pffiffiffi2 is performed on the emitter before the late half of the photon wave packet, corresponding to qubit statej1i, is reflected off the cavity. If the auxiliary spin system is in statej1is(j0is), the photon gets reflected without (with) a π-phase shift in the limit of strong light-matter interaction C≫ 1, charac-terized by the cooperativity C¼ g2=ðκγÞ. Here g is the single-photon Rabi frequency of the cavity-mediated j1i ↔ jei transition, κ is the decay rate of the cavity, and γ is the free-space spontaneous emission rate of the excited level. Up to a global phase, this amounts to a CZ
gate between the photonic time-bin qubit and a qubit in the ground states of the auxiliary spin system initially prepared inðj0isþ j1isÞ=pffiffiffi2. The details of the gate interaction and main errors are described in the Supplemental Material
[26]. The success of the gate is conditioned on sub-sequently detecting the photon in the x basis, which boosts the fidelity. We find that if the intracavity losses are tuned to be on the order of 1=C, then a spin-photon cooperativity of C¼ 100 is sufficient to ensure an error ≲10−4 and a success probability∼0.99ηd, whereηd is the efficiency of
the photon detection.
In the above estimate, we have assumed that reflection of photons into the detector from, e.g., imperfect mode matching is negligible. Such events directly translate into an error since they correspond to operation without any spin-photon interaction. Careful engineering of the mode profile and additional filtering of, e.g., uncoupled polari-zation modes must therefore be employed to suppress such reflections to the desired error level. We note, however, that a mode-matching efficiency of 99% is sufficient to ensure an error of10−4 [26]. Furthermore, we have assumed that the frequency width of the first-level photons is narrow enough to neglect errors from the finite bandwidth of the Purcell-enhanced emitter. For C¼ 100, this would require the first-level photons to have a frequency width∼γ to have errors ≲10−4 [26]. Weak driving from an auxiliary level j2is to jeis [see inset in Fig. 3(a)] allows us to tune the
emission time of the first-level photons to achieve this[27]
(operationM in Fig.4).
We note that depending on the success of the spin-photon Bell measurement, the measurement basis of the qubits in the corresponding branch of the encoded tree-cluster states must be adjusted. If a first-level qubit is lost (detected), the qubits in the corresponding branch should be measured in the x (z) basis for the tree-cluster states generated as described above.
C. Photon measurement
With the considered time-bin encoding, measurements of photons in the z basis require only time-resolved detection. Measurements in the x basis, on the other hand, are more demanding. In particular, a deterministic x-basis measure-ment requires fast optical switching and delay lines. Our analysis shows that gigahertz optical switching rates will be required to ensure tree-generation rates in the megahertz regime (see below). Such switching rates can be exceeded with schemes based on subnanosecond phase control in Mach-Zehnder interferometers via electro-optic modula-tion. Such integrated devices have been demonstrated for a variety of material platforms [28–33]. Toward a scalable, small footprint implementation, we therefore propose an on-chip photonic circuit based on switching via electro-optic modulation in cascaded Mach-Zehnder interferome-ters as shown in Fig.5and discussed in the Supplemental Material, where we also outline an integrated on-chip setup for the repeater stations[26].
D. Other experimental requirements
So far we have discussed the optical interface required to achieve the successful operation of the repeater. In order to be able to make a realistic estimate of the achievable commu-nication rate, we now discuss the concrete requirement for
FIG. 5. Illustration of an on-chip photonic circuit for perform-ing z- and x-basis measurements with integrated sperform-ingle-photon detectors. Mach-Zehnder interferometers with fast switching rate, for example, based on electro-optic switching [28–33], are assumed to switch fully between the top and bottom path (labeled switch) by controlling the relative phase of the two interferometer arms (ϕ). For a z-basis measurement, the photons are guided directly to the detectors for time-resolved recording. For an x-basis measurement, the switch guides the first wave package of the time-bin encoding to the delay line and the second to the short arm. For first-level photons the delay needs to be longer and is replaced with a fiber delay line[26].
two specific physical systems, quantum dots and color centers in diamond. One of the practical requirements in reducing the photon loss is a highly efficient coupling for the cavity to an optical fiber, e.g., using tapered optical fibers
[34–36]. The collection efficiency (β factor) of the emitted photons to a cavity or alternatively a waveguide needs to be high. Quantum dots in waveguides have already demon-strated collection efficiencies of β > 98% [37], which is compatible with the efficiency ηd¼ 0.95 assumed in our
resource analysis below (see Fig.6). In addition, coupling to photonic nanostructures may also decrease the photon emission time through the Purcell enhancement, and photon emission times of ∼100 ps are feasible with solid-state emitters such as quantum dots [38] and diamond color centers[39].
Finally, spin-spinCZgates are required for both the
tree-generation and the reencoding operation. Fast spin-spin gates (∼10 ns) could be performed in stacked quantum dots
[40,41] while somewhat slower gates can be performed between electron and nuclear spins for nitrogen-vacancy
(NV) or silicon-vacancy (SiV) centers in diamond through magnetic dipolar interaction[42]. For the latter, gate times on the order of 100 ns are feasible with SiV systems using nearby nuclei with strong (>1 MHz) hyperfine inter-actions. Alternatively, fast gates could also be implemented using photon-mediated gates between different emitters
[43]. This involves two auxiliary spin qubits for parity measurements and can be made error proof against photon loss errors at the expense of a slight decrease in success probability. For the gate in question [43], a β factor of ∼0.99 would give a success probability of ∼99% and a heralded error of ∼0.1‰. Such probabilisitic spin-spin gates will, however, decrease the rate of the repeater when used in the reencoding step at the repeater stations. The reason being that the reencoding involves the (unprotected) root qubit of the new tree. Considering a distance of 1000 km where ∼400 repeater stations are needed (see optimization below), a success probability of 99% would result in a rate that is∼2% of the rate for a deterministic gate. Using probabilistic gates in the tree generation steps is 0 0 0 1 0 5 5 0 0 1 L [km] 104 105 106 C (a) 0 0 0 1 0 5 5 0 0 1 L [km] 105 106 C (b) 0 0 0 1 0 5 5 0 0 1 L [km] 103 104 105 106 r s [Hz] (c) 0 0 0 1 0 5 5 0 0 1 L [km] 103 104 105 r s [Hz] (d)
FIG. 6. (a),(b) Minimized cost parameter C as a function of distance L assuming spin entangling gate times ofτCZ¼ 10τph(a) and
τCZ¼ 100τph(b) for tree-cluster state generation. The markers correspond to times symbol,ϵr¼ 0.1‰; filled circle, ϵr¼ 0.3‰; circle,
ϵr¼ 0.5‰; square, ϵr¼ 0.1%. Here, ϵr is the error probability of the reencoding operation at the repeater stations. (c),(d)
Corresponding secret bit rate assuming a photon emission time of τph¼ 1 ns and gate time of τCZ¼ 10 ns (c) and τCZ¼ 100 ns
(d). We have restricted the minimization to trees with n≤ 300 photons and repeater station spacings ≥1 km. We have assumed a detection efficiency ofηd¼ 95%. For comparison, the secret bit rate of a two-way repeater (dashed line) with similar resources (see
of less concern since heralding techniques can be employed and the majority of the necessary gates will involve the redundant qubits of the tree encoding, which are somewhat loss tolerant.
The details of the time budget for the generation of the tree-cluster states and length of the necessary delay line is detailed in the Supplemental Material[26]. We find that the photonic tree-cluster states can be emitted within ∼1 μs (∼7 μs) assuming Purcell-enhanced photon emission life-time of about 100 ps[38,39]and spin-spinCZgate times of 10 ns (100 ns). The spin qubits need to stay highly coherent for these timescales, which for SiV and NV systems can be achieved using nearby nuclear spins [44] or operating at low temperatures [45]. For quantum dots, dynamical decoupling [46] or coupling to a nuclear spin memory
[47] may be employed to increase coherence times moti-vating further development of such techniques. For the above generation times, delay lines of maximum length ∼68 m (∼374 m) at the repeater stations are required to ensure the right detection order of the photons (see Sec.III A). At telecom frequencies such a delay line would have a transmission above 99% (∼98%), which can be integrated into the overall detection efficiency ηd.
IV. REPEATER PERFORMANCE
The above analysis outlined all necessary operations and general hardware considerations of the repeater. Importantly, we have shown that only a single successful Bell measure-ment is needed at the reencoding step and that this can be implemented in a loss-tolerant manner using two spin systems. Furthermore, we have outlined how the photonic tree clusters may be generated requiring in total only two qubit spin systems per repeater station in addition to the quantum emitter. We now proceed by estimating the perfor-mance of the repeater in terms of the maximum quantum bit rate for given distances.
The transmission probability of a message qubit through the entire repeater chain will be
ptrans¼ ηmþ1e ; ð4Þ where m is the number of equally spaced repeater stations between the start and end stations, and ηe is the trans-mission probability of the encoded quantum information between repeater stations. The encoded transmission prob-ability depends on the specific tree encoding, the bare transmission probability η of a single photon between repeater stations, and the detection efficiency of the photon detectorsηd. Note that in-out coupling efficiency and any
frequency conversion efficiency that may be required to transduce to the telecom band can be directly included in ηd.
For a tree-cluster encoding with branching vector ⃗t ¼ ½b0; b1;…; bd, ηeis given by the recursive formula[19]
ηe ¼ ½ð1 − μ þ μR1Þb0− ðμR1Þb0ð1 − μ þ μR2Þb1; ð5Þ
where
Rk ¼ 1 − ½1 − ð1 − μÞð1 − μ þ μRkþ2Þbkþ1bk; ð6Þ
with Rdþ1¼ 0, bdþ1¼ 0, and μ ¼ 1 − ηηd. Here, Rk is the probability of having a successful indirect z measure-ment of any given qubit in the kth level of the tree. Consequently, the total probability of a successful z meas-urement of a kth level qubit (direct or indirect) is1−μþμRk. For a fiber-based implementation, the bare transmission will beη ¼ expð−L0=LattÞ, where L0is the distance between the repeater stations and Latt¼ 20 km is the attenuation length of the optical fiber assuming that efficient frequency con-version to the telecom band is implemented.
The relative simplicity of the encoding in tree-cluster states comes with the penalty that it is not able to correct arbitrary errors as opposed to other codes considered for one-way repeaters. It is clear that an error on the qubits participating in the reencoding Bell measurement will map into an error on the encoded message qubit. However, there is some robustness against errors due to the large redun-dancy of information encoded in the tree[14]. As discussed below, this leads to an error rate of the encoded qubits, which is only a few times the error rate of the individual qubits.
To quantify the performance of the tree repeater in the presence of operational errors, we consider the secret bit fraction f of the transmitted qubits, which can be estimated in the asymptotic limit of infinitely long keys assuming perfect classical error correction. Assuming quantum key distribution is performed using a six-state variant of the BB84 protocol we have that[17]
f¼ 1 − hðQÞ − Q − ð1 − QÞh 1 − 3Q=2 1 − Q ; ð7Þ
where Q¼ 2ϵtrans=3 is the qubit error rate of the
trans-mitted bits and hðxÞ ¼ −x log2ðxÞ − ð1 − xÞ log2ð1 − xÞ is the binary entropy. We have assumed a (worst-case) scenario where the noise on the transmitted bits are described by a single qubit depolarizing channel of the form
ΛðˆρÞ ¼ ð1 − ϵtransÞˆρ þ
ϵtrans
3 ðσxˆρσxþ σyˆρσyþ σzˆρσzÞ; ð8Þ where ˆρ is the density matrix of the (pure) message qubit andσx;y;zi are the Pauli matrices. The final error probability of the transmitted message qubit isϵtrans. For a repeater with
m repeater stations, it is estimated as ϵtrans¼ 1 − ð1 − ϵrÞmþ1≈ ðm þ 1Þϵr for mϵr≪ 1, where ϵr is the error
probability of the reencoding step at the repeater stations. Note that f is negative for Q≳ 12.61% reflecting that it is no longer possible to extract any secret bits from the
transmitted qubits since they are too noisy for privacy amplification. Since the tree encoding is not able to correct the errors, this will eventually limit the distance to L∼ 0.13L0=ϵr.
A. Optimization of repeater performance In order to asses the performance of the repeater, we perform a numerical optimization of the number of repeater stations (m) and the encoding tree (⃗t) for a given distance and error (ϵr) to find the highest possible secret bit rate. We
assume that the local repetition rate r0is set by the emission time of the photonic tree-cluster states. For realistic parameters (see below), this will be determined by the emission time of a photonic qubit (τph) and the gate time
(τCZ) of spin-spin gates. For a specific tree encoding
(⃗t ¼ ½b0; b1;…; bd) we estimate the generation time as
1 r0∼b0 100þb1ð1þb2ð1þbd−1ð1þbdÞÞÞ τph þb0 3þb1ð1þb2ð1þbd−2ð1þbd−1ÞÞÞ τCZ:ð9Þ
Note that we assume the emission time of the first-level photons to be100τphto have errors∼10−4in the scattering gate of the reencoding step (see above). In addition, three spin-spin entangling gates are needed for the creation of the first-level qubits. We then seek to minimize the (dimen-sionless) cost parameter,
C¼ 1
r0fptrans
mLatt τphL
; ð10Þ
The inverse cost parameter can be viewed as the secret key rate in units of the photonic qubit emission time per repeater station per attenuation length for a given total distance, L. In the optimization, we enforce a maximum of the number of photons in the encoding of n¼ 300 and require that the repeater stations are never placed closer than 1 km apart. The results of the optimizations are shown in Fig. 6.
It is clear that as the operational errors increase, the repeater performs worse since the tree encoding is not fault tolerant with respect to depolarizing errors. Nonetheless, for ϵr≲ 0.1%, it is still possible to reach high secret bit
rates since the repetition rate is determined by the local repetition rate, which can be in the megahertz regime. Specifically, with a photon emission time ofτph¼ 1 ns and
gate time ofτCZ¼ 10 ns, a secret bit rate of ∼70 kHz over
1000 km is possible with a repeater station spacing of 2.6 km, a detection efficiency ofηd¼ 95%, a reencoding
error of ϵr¼ 0.3‰, and using ½4; 14; 4 trees of 285
photons. For more modest gate times of τCZ ¼ 100 ns, a
secret bit rate of ∼13 kHz over 1000 km for the same parameters can be achieved (see above and Supplemental
Material [26] for a justification of these numbers for a concrete physical realization).
B. Logical errors
We have assumed a generic reencoding error ϵr in our optimization above. This encoding error will, in general, be determined by errors from both the generation of the photonic tree clusters and the reencoding step. In the optimization, we assumed a fixedϵrand optimized the tree encoding for the given distances (see Supplemental Material [26]). One could imagine that ϵr will depend on the size of the encoding. To investigate this, we consider single qubit depolarizing channels of the form in Eq.(8)
acting on all qubits in the encoding. We can then estimate the single qubit error probabilityϵ that will result in a given reencoding error probabilityϵrfor a specific tree encoding as detailed in the Supplemental Material [26]. The tree encodings are remarkably robust to errors even in the presence of loss and we find that ϵr=ϵ ≈ 3 for tree encodings and loss corresponding to the optimization in Fig.6except for the high error (ϵr¼ 0.1%) optimization, where we find thatϵr=ϵ ≈ 4–5. Notably, we do not find any significant dependence of ϵr on the different tree encod-ings. This is consistent with the errors of the two qubits (first-level qubit and root qubit) participating in the Bell measurement dominating the reencoding error.
C. Comparison to other approaches
The proposed repeater compares favorably to previously proposed one-way repeater protocols [12–14,16,18] (see Table 1 in the Supplemental Material [26]). It enables similar secret key rates for roughly the same error param-eters and detection efficiencies. The key advantage of this repeater, however, is that it requires substantially less resources per repeater station than any of the previous protocols. In particular, the number of spin qubits per repeater station is 2 orders of magnitude lower than the matter based protocol in Ref.[12]and the large overhead of single-photon sources for linear optics protocols [14] is circumvented. The resources for the latter may be reduced by generating the photon cluster states as proposed in Ref.[21]. Nonetheless, the size of the encoding is still more than an order of magnitude larger than for our protocol. We obtain this reduction by directly encoding the message qubit in a tree-cluster state and using the spin-photon interface for near-deterministic reencoding operations as opposed to swapping entanglement with probabilistic linear optics Bell measurements. Compared to other one-way repeaters, the proposed repeater is, however, not fault tolerant and the tolerable error level therefore decreases with the distance. This is simply a consequence of the buildup of error with each reencoding operation. Thus, and order of magnitude decrease in error roughly corresponds to an order of magnitude increase in distance (e.g., distances in the range 103–104 km would be achievable
with an error rate of 10−4–10−5 and tree-cluster states of depth 3). It might be possible to remedy this effect by incorporating error correcting for logical errors at the expense of a few additional spins at each station. A full investigation of this is, however, beyond the scope of this article.
We have also compared the repeater to a two-way quantum repeater allowing for parallel entanglement gen-eration attempts using the same total number of spin qubits as our one-way repeater (see Supplemental Material[26]). We have assumed deterministic noise-free entanglement swapping and noise-free entanglement generation using a two-photon interference scheme[48]. This provides a crude comparison with standard two-way repeaters. As shown in Fig.6(b), our one-way repeater reaches key rates orders of magnitude higher than the two-way repeater due to the higher local repetition rate. Furthermore, this advantage is achieved without the need for long coherence times. For the two-way repeater, orders of magnitude larger coherence times (miliseconds to seconds) are required. Note that Ref.[7]contains an extended comparison between one-way and two-way repeaters also showing the advantage of the former in the low noise limit. The comparison performed so far has been made in terms of communication rate per qubit by requiring the two repeater approaches to have the same total number of qubits. In practice, other factors may also be relevant for the comparison. In particular, the number of repeater stations is about an order of magnitude larger for the one-way repeater and the initial cost of establishing a quantum repeater chain is thus likely to be higher with the present approach. This is, however, compensated by a much higher communication rate resulting in a better rate per qubit.
V. CONCLUSION AND DISCUSSION
We have proposed a novel one-way quantum repeater based on photonic tree-cluster states. The repeater enables secret bit rates ∼70 kHz (∼13 kHz) over a distance of 1000 km assuming gigahertz single-photon emission rates and spin-spin entangling gate times of 10 ns (100 ns). We have demonstrated how both the error correction and the generation of the tree-cluster states at the repeater stations can be performed with a minimum number of spin systems. Specifically, we have outlined a repeater setup that requires only a single quantum emitter and two memory spin qubits per station. As compared to the daunting requirement for realizing previously proposed one-way quantum repeaters, this places our proposal within experimental reach of current technologies.
Solid-state systems such as quantum dots and diamond defects are promising hardware candidates. Single-photon emission rates exceeding gigahertz have already been achieved[38,39]together with efficient coupling to nano-photonic waveguides and cavities [42,49–52]. The spin-spin gates required for the repeater may be mediated
through tunneling in quantum dots [40,53,54] or nuclear-electron spin coupling for diamond defects [42]. While many of the key elements necessary for this proposal have already been demonstrated, additional engineering of the platforms will be required in order to reach the required photon collection efficiencies and error level of the gates. Importantly, the proposed implementations based on state-of-the-art solid-state emitters appear capable of reaching those demanding metrics. Notably, both high cooperativity and combined detection and in-out coupling efficiencies above 90% have recently been reported with a SiV defect center coupled to a nanophotonic cavity[55]. For quantum dots, chip-to-fiber coupling efficiencies exceeding 80% have been reported[36], which could readily be improved further. High cooperativity has also recently been demon-strated by coupling a quantum dot to a microcavity[56]. The current state of the art is thus not far from the required performance of this protocol. In addition, our protocol outperforms direct transmission already at overall detection efficiencies of 85%, and for slightly higher efficiencies, error levels >0.1% can be tolerated, as shown in the Supplemental Material[26]. Both efficient frequency con-version to the telecom C band (where low-loss optical fibers exist) and fast optical switching are necessary to achieve long communication distances. While this remains a challenge, there has been impressive progress on high-efficiency frequency conversion [57–59] and fast optical switching[60]with clear routes toward further improving the performance. An alternative strategy is to develop quantum dots emitting directly in the telecom C band
[61], which relies on the continuous development of material growth technology. It thus seems feasible that our proposal provides a promising experimental route toward high-rate quantum key distribution with potential for proof-of-principle experiments with current technology.
ACKNOWLEDGMENTS
We would like to thank Dirk Englund, Matthias Christandl, Martin Hayhurst Appel, Ralf Riedinger, and Mihir Bhaskar for many valuable discussions. P. L. and A. S. S. gratefully acknowledge financial support from the Danish National Research Foundation (Center of Excellence “Hy-Q,” Grant No. DNRF139), the European Research Council (ERC Advanced Grant“SCALE”), and the European Union’s Horizon 2020 research and innova-tion programme under Grant Agreement No. 820445 and project name Quantum Internet Alliance. J. B. acknowl-edges financial support from VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059). T. S. was supported by the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska Curie Grant Agreement No. 753067 (OPHOCS) and the Federal Ministry of Education and Research of Germany (BMBF, Project No. DiNOQuant13N14921). Work at Harvard was
supported by NSF, CUA, NSF EFRI, ARL, and Vannevar Bush Fellowship.
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